Over the past half-century, the tools of neuroscience have revealed much about the workings of the human brain. Now researchers are pushing forward in a new frontier: exploring what goes on in the mind of man’s best friend. The study of canine cognition has taken off in recent years, energized by new findings about how dogs learn words, numbers and abstract concepts — and how they manage us, their ostensible masters.

In a study published in the journal Animal Cognition, researchers used a procedure known as “preferential viewing” to show that dogs can understand simple calculations. Eleven pet dogs were shown treats that were then placed behind a screen. When one treat, and then another, was placed behind the screen and the screen was removed, dogs gazed briefly at the two treats. When two treats were deposited behind the screen but only one remained when the screen was taken away, the dogs stared at the lone treat for longer, indicating that they were aware the math didn’t add up.

Dogs understand language too, and the new research shows they can learn more words than just down and sit. The average dog can learn 165 words, notes psychologist Stanley Coren of the University of British Columbia, and some superdogs can have a vocabulary of 250 words. In a study that appeared in the journal Science, researchers at the Max Planck Institute for Evolutionary Anthropology in Germany reported on a border collie named Rico who could learn a name given to an unfamiliar object like a stuffed bunny through a process of elimination and could remember the name of that object four weeks later. (Border collies lead the list of the most intelligent dog breeds, according to a survey of dog-obedience judges; they’re followed by poodles, German shepherds, golden retrievers, Dobermans, Shetland sheepdogs and Labrador retrievers.) Dogs can learn to solve spatial problems — figuring out the fastest route to a favorite chair, locating a hidden treat — and can learn to operate simple mechanisms like latches.

Most impressive of all is dogs’ ability to learn about humans. They respond to our gestures, they attend to our body language, and they follow our gaze to figure out what we’re looking at. They even are susceptible to repeating human yawns, according to a study published in the journal Biology Letters. As the longest-domesticated species, dogs have evolved alongside humans, selected over thousands of years for traits that make them especially sensitive to our cues. Another study from the journal Science reported that puppies only a few weeks old could interpret human signals, while full-grown wolves raised by humans could not. Dogs read people better than do chimpanzees, humans’ closest primate relative, according to research published this year. In fact, the most accurate comparison is to a human child: dogs have the social-cognition capacities of a 2-year-old. (The dogs in one recent study can claim another similarity to iPhone-loving toddlers: their ability to understand abstract concepts was probed by having them use touchscreen computers.)

Even deeper insight into the canine mind may be on the way. Scientists this year reported the results of the first brain scans conducted on awake, unrestrained dogs that were trained to lie perfectly still inside an MRI machine. The aim of the experiment was to find out which brain circuits would respond when the dogs’ human owners made a gesture offering food. The scans showed that when a treat was promised, a pattern of activity appeared in the caudate nucleus, a part of the brain associated with the anticipation of reward. (The dogs wore special earmuffs to protect their hearing in the noisy machine, and one dog in particular, a former shelter dog named Callie, enjoyed the approval she got from experimenters so much that she would jump into the scanner uninvited.)

Gregory Berns, the Emory University researcher who led the MRI study, writes that there are “endless” questions still to be explored: How do dogs distinguish among the humans they know; is it by sight or smell? What meaning does our language have to them? How do their minds represent humans and other animals, including other dogs? The study of canine cognition, he notes, ultimately brings us back to our own desires and behaviors: “Because humans, in effect, created dogs through domestication, the canine mind reflects back to us how we see ourselves through the eyes, ears, and noses of another species.”